Use the table of integrals to find:
∫ x³*e⁹ˣ dx
so, did you use the table? It will probably provide a reduction formula, such as
∫x^n e^ax dx = 1/a x^n e^ax - n/a ∫ x^(n-1) e^ax dx
this is achieved using integration by parts. So, going through that three times gives you
(1/9 x^3 - 1/27 x^2 + 2/243 x - 2/2187) e^9x + C
I came across this young man with his remarkable youtube math movies
called "blackpenredpen".
He shows a short-cut method to Integration By Parts calling it the DI method,
also called the Tabular Method. He uses 3 different ways in which the process
will end.
It works really well here:
https://www.youtube.com/watch?v=2I-_SV8cwsw&t=695s
∫ x³*e⁹ˣ dx = (1/9)x^3 e^(9x) - (1/27)x^2 e^(9x) + (2/243)x e^(9x) - (6/6561) e^(9x)
= (1/9)e^(9x) ( x^3 - (1/3)x^2 + (2/27)x - 2/243) + c
---D ----- I
+ | x^3 -- e^(9x)
- | 3x^2 -- (1/9)e^(9x)
+ | 6x ----- (1/81)e^(9x)
- | 6 ------- (1/729)e^(9x)
+ | 0 ------ (1/6561)e^(9x) <----- stop when you reach a zero in the D column, multiply the diagonals.
To find the integral of x³ * e⁹ˣ, we can use the table of integrals. In this case, we need to use the formula:
∫ x^n * e^px dx = (x^n / p) * e^px - (n/p) * ∫ x^(n-1) * e^px dx
Using this formula, we can determine the integral step by step.
Step 1: Identify the values of n and p in our integral.
- For our integral ∫ x³ * e⁹ˣ dx, n = 3 and p = 9.
Step 2: Apply the formula to the integral.
∫ x³ * e⁹ˣ dx = (x³ / 9) * e⁹ˣ - (3/9) * ∫ x² * e⁹ˣ dx
Step 3: Simplify the remaining integral and apply the formula again.
- The second term, ∫ x² * e⁹ˣ dx, can be further simplified using the same formula:
∫ x² * e⁹ˣ dx = (x² / 9) * e⁹ˣ - (2/9) * ∫ x * e⁹ˣ dx
Step 4: Simplify the remaining integral and apply the formula once more.
- The third term, ∫ x * e⁹ˣ dx, can be simplified using the formula:
∫ x * e⁹ˣ dx = (x / 9) * e⁹ˣ - (1/9) * ∫ e⁹ˣ dx
Step 5: Integrate the final integral.
- The fourth term, ∫ e⁹ˣ dx, is simply e⁹ˣ.
Step 6: Substitute the results back into the original integral.
∫ x³ * e⁹ˣ dx = (x³ / 9) * e⁹ˣ - (3/9) * [(x² / 9) * e⁹ˣ - (2/9) *[(x / 9) * e⁹ˣ - (1/9) * e⁹ˣ]]
Step 7: Simplify the expression and combine like terms, if necessary.
So, the final result of the integral ∫ x³ * e⁹ˣ dx is:
(x³ / 9) * e⁹ˣ - (3/9) * [(x² / 9) * e⁹ˣ - (2/9) *[(x / 9) * e⁹ˣ - (1/9) * e⁹ˣ]]